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Theorem bj-clel3gALT 34844
Description: Alternate proof of clel3g 3558. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-clel3gALT (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-clel3gALT
StepHypRef Expression
1 elisset 2814 . . . 4 (𝐵𝑉 → ∃𝑥 𝑥 = 𝐵)
21biantrurd 536 . . 3 (𝐵𝑉 → (𝐴𝐵 ↔ (∃𝑥 𝑥 = 𝐵𝐴𝐵)))
3 19.41v 1957 . . 3 (∃𝑥(𝑥 = 𝐵𝐴𝐵) ↔ (∃𝑥 𝑥 = 𝐵𝐴𝐵))
42, 3bitr4di 292 . 2 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝐵)))
5 eleq2 2821 . . . . 5 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
65bicomd 226 . . . 4 (𝑥 = 𝐵 → (𝐴𝐵𝐴𝑥))
76pm5.32i 578 . . 3 ((𝑥 = 𝐵𝐴𝐵) ↔ (𝑥 = 𝐵𝐴𝑥))
87exbii 1854 . 2 (∃𝑥(𝑥 = 𝐵𝐴𝐵) ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥))
94, 8bitrdi 290 1 (𝐵𝑉 → (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐵𝐴𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811
This theorem is referenced by: (None)
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